1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove that the set {X: XA = AX} is a subspace of M 2,2

  1. Mar 10, 2012 #1
    1. The problem statement, all variables and given/known data
    A is a 2 x 2 matrix. Prove that the set W = {X: XA = AX} is a subspace of M2,2


    2. Relevant equations



    3. The attempt at a solution
    I have already proven non-emptiness and vector addition.

    Non-emptiness:

    Code (Text):
    W must be non-empty because the identity matrix I is an element of W.
    IA = AI
    A = A
    Vector addition:

    Code (Text):
    Let X, Y be elements of W such that XA = AX, YA = AY. Add the equations together.
    (X+Y)A = A(X+Y)
    XA+YA = AX+AY

    Hence, X+Y is an element of W.
    I have no idea what to do with scalar multiplication. This is my current attempt:

    Code (Text):
    Let X be an element of W, and c be a scalar.
    cXA = AcX
    (cX)A = A(cX)
    I also tried this:

    Code (Text):
    Let X be an element of W, and c be a scalar.
    XA = AX
    c(XA) = c(AX)
    I don't feel like either of these attempts proves anything.

    I don't understand how to prove that cX is actually in W. This doesn't make any sense to me. How do I prove that?
     
  2. jcsd
  3. Mar 10, 2012 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    It's too trivial.
    You have X in W, thus XA=AX, thus cXA=A(cX) meaning cX in W.
    Or in other word you've proved it already.
     
  4. Mar 10, 2012 #3
    But how do I know that there isn't a scalar that would change X in such a way that (cX)A ≠ A(cX)?

    Edit: I guess I get what you're saying now, but it still confuses me how I can just say that they're equal with no real evidence.
     
  5. Mar 10, 2012 #4

    MathematicalPhysicist

    User Avatar
    Gold Member

    They are equal cause you have XA=AX, and then you multiply this equation by c.
    I said it's trivial.
     
  6. Mar 10, 2012 #5

    MathematicalPhysicist

    User Avatar
    Gold Member

    Remember that your scalar is some number, real/complex or something else.
    It's not a vector (in which case multiplication of a matrix with a vector will not yield you a matrix of the same size but a vector, and thus it wouldn't be a vector space.
     
  7. Mar 15, 2012 #6
    Yeah, I understand it now. This is the first class I've had to write my own proofs in, so I'm still getting used to it. Thanks again for the help.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Prove that the set {X: XA = AX} is a subspace of M 2,2
  1. Var(ax^2 - x ) (Replies: 7)

Loading...